Harmony of Numbers

You don’t have to be a musician to know when music sounds bad. But what is it, exactly, that makes music sound good? Turns out it’s all in the numbers. The combinations of notes we find pleasing are based on simple ratios.

In this lesson, students learn how the notes plucked on stringed instruments are mathematically related. By calculating the ratios of string lengths, students explore the relationship between notes that sound good together — and also those that don’t. Math and music have more in common than you might think!

Students will

Measure string lengths and calculate the ratios for different notes in a major scale

Apply fraction division to calculate the length ratio between any pair of strings

Explore the relationship among string lengths, ratios, and sound quality of simple harmonies

Before you begin

This lesson could be used to introduce the concept of ratios and explore their relationship to fractions, or could serve as an application of ratios. Either way, students should have a strong foundation in fractions. They should be comfortable multiplying fractions by whole numbers and other fractions, and some previous exposure to fraction division would also be helpful.
Although the lesson focuses on music, no prior musical knowledge is assumed, either for the student or the teacher!

How do the rules of an election affect who wins? Students calculate (as a percent) how much of the electoral and popular vote different presidential candidates have received, and add with integers to explore elections under possible alternative voting systems.

Topic:
Number System (NS), Ratios and Proportional Relationships (RP), Reasoning with Equations and Inequalities (REI)

Is Wheel of Fortune rigged? Students use percents and probabilities to compare theoretical versus experimental probabilities, and explore whether the show is legit, or whether there might be something shady going on!

How many calories does a body burn? Students interpret and apply the formula for resting metabolic rate (RMR) in order to learn about how calories consumed from food, calories burned from exercise, and calories burned automatically contribute to a body's weight.

How much of your life do you spend doing different activities? Students use proportional reasoning and unit rates to calculate how much of their total lifespan they can expect to spend sleeping, eating, and working...and discuss how they'd like to spend the time that's left over.

When you buy a concert ticket, where does your money go? Students use percents and proportional reasoning to describe how revenue from tickets is distributed among the various players in the concert game.

What size ice cubes should you put in your drink? Students use surface area, volume, and rates to explore the relationship between the size of ice cubes and how good they are at doing their job: chilling.

How much should people pay for cable? Students interpret scatterplots and calculate the costs and revenues for consumers and providers under both the bundled and à la carte pricing schemes to determine which would be better for U.S. companies and customers.

How much do different professionals earn in a year? Students use rates and ratio reasoning to compare how much a teacher, the President, and LeBron James earn...and to compare how much value the create.

How does the what we see affect our happiness? Students explore the concept of the jen ratio – the ratio of positive to negative observations in our daily lives – and use it to discuss how the content we consume and the things we observe influence our experience of the world.

Like the jacket, this lesson is for Members only.

Some Free Lessons From Mathalicious

How do the rules of an election affect who wins? Students calculate (as a percent) how much of the electoral and popular vote different presidential candidates have received, and add with integers to explore elections under possible alternative voting systems.

Topic:
Number System (NS), Ratios and Proportional Relationships (RP), Reasoning with Equations and Inequalities (REI)